Ponder This

December 1999

Millennial Puzzle
The puzzlemaster misstated the December puzzle the first time around.
Let's try again.

Consider the integer N=2^1999 (2 raised to the 1999 power).
Is there a positive integer multiple of N whose decimal representation
does not contain the digit 0?
How would you construct such an integer or prove that it doesn't exist?

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